\[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x} \, dx \]
Optimal antiderivative \[ -\frac {a \cosineIntegral \left (\frac {3 f x}{2}\right ) \cos \left (\frac {3 e}{2}+\frac {\pi }{4}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{2}+\frac {3 a \cos \left (\frac {e}{2}+\frac {\pi }{4}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sinIntegral \left (\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{2}+\frac {a \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sinIntegral \left (\frac {3 f x}{2}\right ) \sin \left (\frac {3 e}{2}+\frac {\pi }{4}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{2}+\frac {3 a \cosineIntegral \left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{2} \]
command
integrate((a+a*sin(f*x+e))^(3/2)/x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (a f \cos \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) \operatorname {Ci}\left (\frac {3}{2} \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, a f \cos \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) \operatorname {Ci}\left (\frac {1}{2} \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + a f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {3}{4} \, \pi - \frac {3}{2} \, e\right ) \operatorname {Si}\left (\frac {3}{2} \, f x\right ) + 3 \, a f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {1}{4} \, \pi - \frac {1}{2} \, e\right ) \operatorname {Si}\left (\frac {1}{2} \, f x\right )\right )} \sqrt {a}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________